Here's a discussion you don't quite see everyday. Fan of this great film and related media materials.

The visuals and depiction during these scenes are somewhat ambiguous, however I believe the intention was definitely that the Klingons as well as Enterprise encountered/confronted the Vejur cloud at warp velocities.

Kirk indicates to Scotty that the Vejur cloud is "less than 3 days away from this planet" at the time of Epsilon 9's data transmissions alerting Starfleet of the approaching cloud. It would not be possible for Vejur to reach Earth from the Klingon/Federation border at sub-light velocities.

The novel though not official canon, elaborates greatly on the notion of how the Klingons and Enterprise achieved approaching the "intruder" cloud. The cloud was stated as traveling at Warp 7 in the novel if memory serves. At that velocity the cloud passed through Klingon space so rapidly, that particular formation of cruisers were the only ships available to intercept. They simply happened to be on patrol of the border. I believe the intention was that the cloud did not traverse the entirety of Klingon space but rather popped in from another region and skirted the Klingons frontier sectors.
The Klingons and Enterprise would have overtaken the cloud from behind as it traveled at warp velocities. This would infact give the appearance as they approached of both objects traveling at sublight velocities as they maintained a constant velocity relative to each other. Its entirely possible though no evidence to suggest, that the cloud was of such enormous power it would have been possible to infact disengage warp once the power field was penetrated and still be "carried" along with the vessel at warp velocities, much as our own heliosphere around our solar system acts as its own self contained enviroment from instellar space.
The Klingons of course possessed the same scanning technology as Epsilon 9 and the Enterprise. They simply reacted with hubris to the violation of their space by an unknown "intruder." Being a war-like species this hubris blinded them to the threat a mere "cloud" could pose. Once their weapons were rendered ineffective and the first cruiser was "digitized", the Klingons then retreated. One could argue they simply could have dropped out of warp and immediately been millions upon millions of miles away from harm as the cloud and plasma energy weapon discharges sped away at warp velocities. No doubt there was a certain amount of panic involved when the first cruiser was reduced to data. It should be noted it is a true testament to the Enteprise technology and shields to survive even one direct impact of the plasma bolt. The Klingon starships are built for combat and warfare and their deflector screens could not absorb the plasma energy. It was said Vejur has knowledge that spans the entire universe, and Spock witnessed not only Epsilon 9, but planets, moons, stars, even entire galaxies that had been digitized and reduced to data by Vejur. In the novel, it indicates Vejurs curiosity was piqued when the Enterprise wasn't initially digitized by the offending plasma bolt. Poor Epsilon 9 didn't know what hit it.